Computational Approaches to Stochastic Vehicle Routing

Problems

Dimitris Bertsimas Philippe Chervit Michael Peterson

April 18, 1991

Abstract

We develop and test several heuristics for two well-known vehicle routing problems which

have stochastic demand: the probabilistic traveling salesman problem (PTSP) and the prob-

abilistic vehicle routing problem (PVRP). The goal in these problems is to obtain optimal a

priori tours, that is, tours which are designed before demands are known. We develop a number

of heuristics which lead to such solutions, examine the quality of these solutions, and perform

computational experiments against alternative strategies of re-optimization, which we are able

to compute using sampling techniques. Based on extensive testing we conclude that our heuris-

tics are very close in terms of performance compared with re-optimization strategies and can be

used by a practitioner.

*D. Bertsimas, Sloan School of Management, MIT. The research of D. Bertsirnas was partially supported by the

National Science Foundation research initiation grant DDM-9010332 and NSF grant DDM-9014751.

t P. Chervi, Operations Research Center, MIT.tM. Peterson, Sloan School of Management, MIT. The research of M. Peterson was partially supported by NFS

graduate fellowship grants RCD-88-54827 and RCD-90-50002

1

1 Introduction

This paper examines two well-known and related problems in optimization under uncertainty: the

probabilistic traveling salesman problem (PTSP) and the 1-vehicle probablistic vehicle routing prob-

lem (PVRP). Below we give formal definitions of both problems, but more simply, these problems

can be thought of as identical to their deterministic counterparts except that the demands of the

points to be visited are random rather than fixed.

We shall be concerned throughout the paper with comparing what we call a priori strategies to re-

optimization strategies. An a priori strategy is one which is formulated in advance of any information

about the demands (other than their locations). In contrast, a strategy of re-optimization allows

tours to be designed after demand information is known, so that each re-optimized solution is tailored

to a particular realization. Of course, re-optimization, where practical, will give better solutions than

a priori strategies. However, in many practical situations, such a strategy might prove prohibitively

costly or may not be desirable for other reasons. For example, a delivery company may wish to

have its drivers follow the same route each day in order to gain familiarity with a particular set

of customers. Moreover, as we shall see, re-optimization solutions may have resource requirements

entirely too high for smaller firms, firms which might nevertheless wish to improve operational

performance.

Let us define the two problems in which we are interested. Consider a graph G(V, E) on n

nodes representing locations to be visited. An instance of the PTSP is specified by a matrix D of

inter-node distances together with a set of probabilities {Pi}, each pi specifying the probability that

a given node i must be visited on a realization of the problem. Such a node is said to be present for

that realization. In the case where pi = p for all i we have an instance of the homogeneous PTSP;

otherwise we have a heterogeneous PTSP. The goal is to find an a priori tour r (i.e. a tour designed

prior to any realization) to minimize the expected length

E[LT] E p(S)L(S), (1)scv

where each Lr(S) is a tour visiting the nodes present on a given realization S in the same order as

they appear in the tour r. The term p(S) is the probability of realization S and is given by

p(S) = Pi II (1 - Pi). (2)iEs iqs

An alternative to the strategy of designing an a priori tour is that of re-optimization, where for each

2

realization S, an optimal TSP tour is re-calculated. The cost under re-optimization is given by

E[ETSP] = E P(S)LTSP(S), (3)scv

where LTSP(S) is the length of the optimal traveling salesman tour through the points of S. Al-

though D may be general, in this study we focus on Euclidean distances only.

Some simple additions to the PTSP just described give us the 1-vehicle PVRP, the second

problem to be examined. Consider a set of (random) demands at the n locations, a point "O"

which we designate as the depot, and a single vehicle of capacity Q. We suppose that the vehicle

must collect loads from the n locations, returning periodically to a depot to empty. The demands

X 1,.. , Xn, are independent integer random variables, each with a known probability distribution,

and the objective is to design a routing strategy to minimize expected travel distance. In order to

define the problem more precisely we introduce the notation:

Q vehicle capacity

K = maximum demand value, K < Q

d(i, j) _ distance between nodes i and j

s(i, j) - d(i, O) + d(O, j) - d(i,j)

= Pr{demand at node i equals j}

f(m,r) - Pr{demand for 1,...,m is r}

As with the PTSP, a strategy of re-optimization is available, though in practical situations it might

be prohibitively expensive.

Two cases (which we call a and b) arise in the PRVP. In case a, the designed routes are followed

exactly, so that all locations (even those having zero demand on the day) are visited. In case b,

the same routes are used, but locations having zero demand are simply skipped. In both cases, the

total expected distance traveled will be that of the a priori route chosen plus certain radial terms

stemming from returns to the depot. More formally, let 7i denote the probability that the vehicle

exactly reaches its capacity at node i, and i the probability that the vehicle exceeds its capacity at

node i. Using our f(m, r) notation these probabilities are given by

71 = 0 (4)

i E Epif(i - l, q Q - k ) , 2<i<n, (5)q=l k=1

3

(6)L*J K-1 K

= z z z P. f(i-1 qQ-k) 2 < i < n, (7)=l k=l r=k+l

The probabilities f(m, r) can be computed from the recursion

min(K,r)

f(m,r)= Pk yf(m -1,r-k),m > 2,r < Km (8)k=O

with the initial condition

f(l,r)=p7', O<r<K. ()

Now re-number the points 1... n so that point 1 is the first point on our a priori tour, point 2 is

second, etc. Expected tour lengths are then given by

n n

E[L a] = a d(i, i + 1) + Z(bis(i, i) + ys (i, i + 1)) (10)i=O i=1

n i-1 n n

E[L] = Ed(O,i)pi p + d(i,O)pi 11 p +i=1 r=l i=1 r=i+l

n n j-1

a, Z d(i,j)pipj Il P +

i=1j=i+l r=i+l

E is(i,i) + 7ipjs(ij) I ' Pr (11)i=1 j=i+l r=i+l

where Pi = 1- p, the probability of positive demand at node i, and n + 1 = 0, the depot node. The

first expression consists of the underlying tour plus radial correction terms for returns to the depot.

These correction terms vary according to whether capacity is exactly met or strictly exceeded. The

second expression derives from the same principles, but is more complicated since zero demands may

be skipped.

The PTSP was introduced and analyzed by Jaillet [13] and further explored by Bertsimas and

Howell [6]. The PVRP with general demand distributions was analyzed in Bertsimas [7]. For a

comprehensive survey of the area of a priori optimization see [8].

In this paper we undertake the task of performing extensive computational testing of several

heuristic procedures. In particular we address the following issues

* How do simple heuristics compare to re-optimization?

* At what problem size does asymptotic behavior emerge?

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11

* What are the relative effects of more sophisticated heuristics like local improvement and dy-

namic programming procedures?

* What are the relevant conclusions for the practitioner?

A novel aspect of our approach is the use of sampling techniques combined with variance reduction

techniques in order to get more precise estimates. In this way we are able to compare a priori and

re-optimization strategies.

All of our subsequent analysis will deal with proposing and analyzing a priori heuristic approaches

for these two problems. Sections 2 and 3 do this for the PTSP, while sections 4 and 5 do this for

the PVRP. Section 6 summarizes the conclusions of the paper.

2 Heuristics for the Probabilistic Traveling Salesman Prob-

lem

In this section we describe the heuristic methods we apply to the PTSP. These fall into two categories:

tour construction techniques (these are typically used as the first step in our procedures), and local

improvement procedures. For a more detailed discussion of the procedures, consult the references

given.

2.1 The Space-filling Curve Heuristic

Spacefilling curves, first introduced by Peano, Hilbert, and Sirpinski, have more recently come

into use in solving routing and partitioning problems (c.f. Bartholdi and Platzman [2], [3]). Our

description here will omit most technical details.

We define a spacefilling curve to be a continuous mapping IF : C -a S where C is the unit

interval and S is the unit square. The specific form of 'I will not be given here, but it has several

important properties:

* is onto but not necessarily surjective. Thus any image in S has one or more pre-images in

C.

* A pre-image 0 satisfying P(8G) = (, y) may be computed in O(k) operations, where k is the

number of bits needed to specify the image (z, y).

5

* There is a function f on C with f(O) = 0, f(0) = f(1 - 0), such that

II () - (6') II< f(l 0 - 6' ) (12)

where I1 11 is our metric on S.

It is the third property listed above which gives the spacefilling curve its usefulness in the context

of the TSP. Given our Euclidean problem in the unit square, we transform the points to the unit

interval via our function and construct a TST based on the order of the nodes in this interval (i.e.

we sort the points of an arbitrary starting tour according to their location under the mapping AI, then

recover a new tour by finding the pre-images). The third property above is known as the property of

closeness. In simple terms, it means that points which are close in the square are mapped to points

which are close in the unit interval (and thus are near each other in the tour so constructed). In the

worst case, the spacefilling curve algorithm has a performance guarantee of

H L < O(logn), (13)

where Hn is the length obtained by the heuristic and Ln is the length of the optimal TST. This result

is due to Steele [14]; Bertsimas and Grigni [5] show this bound is tight. Computational analysis

shows that the algorithm differs by 25% from optimal on average, so in our studies we supplement

it with local improvement procedures.

2.2 The Radial Sorting Heuristic

The radial sorting heuristic takes as an origin the center of mass of the n points, then visits the points

in order of their radial angle from this origin. The resulting star-shaped tours give low expected

cost in the case of the homogeneous PTSP for low values of p, the intuition being that for such

low values, most points will not be present and the resulting tour will have the desirable property

of convexity. As with the spacefilling curve, we supplement radial sorting with local improvement

procedures.

2.3 The 2-interchange: Homogeneous PTSP

The 2-interchange local improvement heuristic for the PTSP is essentially an adaptation of that

used in the TSP. The procedure takes an existing tour and tries all possible interchanges of pairs of

nodes until there is no further improvement in the expected cost. Here we follow the presentation

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i-i i-i 1~· 1-I I

-1

- I 4j+I j+l jw j

Figure 1: Illustration of the two-interchange procedure. At top is original configuration. In cen-

ter is the configuration I(i,j) obtained by interchanging i and j. At bottom is the configuration

I(i + 1,j - 1) obtained by interchanging i + 1 and j - 1.

of Bertsimas and Howell [6] in demonstrating the procedure for the case of the homogeneous PTSP;

some details for the more complicated heterogeneous case are given in the next section.

Suppose we have an ordering of points

(1,...,i- 1, i,i + 1,...,j- 1,j,j + ,...,n).

Inversion of points i and j gives the new order I(i, j):

(1,...,i- 1,j,j- 1,...,i + 1,i,j + 1,...,n).

The heuristic operates by computing the difference in cost between two configurations obtained from

one another via such interchanges. We compute here the cost difference between the configurations

I(i,j) and I(i+ 1,j-1). Let C(i,j) represent the cost of I(i,j). We shall call nodes i+1, ... , j -1 the

inside nodes and all others for this configuration the outside nodes (refer to Figure 1). The differences

between one configuration and another are a function of "interactions" between the nodes i, j and

the sets of inside and outside nodes. Defining the arrays A and B according to

n-l

Ai,k = Z(1-p)"-ld(i,i+ r)r=k

n-1

Bi,k = Z(1-p)r-ld(i, i-r),r=k

we have the formula

C(i,j) = C(i + 1,j -1) +

7

i- i-i i

d

!

p2 [(q-k _ 1)Ai,k+l + (qk -_ 1)(Bi, 1 - Bi,n-k)

(qk - 1)(Aj,l - Aj,.-k) + (q-k _ 1)Bj,L+1 +

(q'-k _ 1)(Ai,1 - Ai,k) + (qk-f - 1)Bi,n-k+l +

(qk-n - 1)Aji,n-k+ + (q-k - 1)(Bj,1 - Bj,k+l)] , (14)

where k = j - i, q = 1 - p, and all indices are taken modulo n. Each line of this formula represents

a different group of interactions. For example, the first term concerns the interaction between node

i and the set of outside nodes; it may be derived as follows. For the configuration I(i,j), the

contribution to the total expected cost is

n-1 n-k-1[p2 E d(i, i + r)(1-p)r t + E d(i, i-r)(1-p)r 1 (15)

r=k+l r=1

The term p2 appears because nodes i and i + r (or i - r) must be present, while the requirement

that the other nodes be absent gives the apppriate 1-p term. The corresponding expression for the

configuration I(i + 1,j - 1) is

n-1 n-k-1

p2 d(i, i + r)(1-p)- '+ d(i,i-r)(1-p) +k . (16)=rk+l r=l

Subtraction of this expression from (15) gives

n-1 n-k-1p2 d(i + r)(1- p)rl(q-k _ 1)+ d(i, - r)(1 - p)r-l(qk 1)

r=k+l r=1

which, upon substitution of our A and B notation, becomes the first line of the formula in (14).

In a similar fashion to the above, we can obtain the second line of the formula from the interaction

between i and the inside nodes, and the third and fourth lines from the corresponding interactions

for node j. There are no contributions from other interactions due to the symmetry of the problem

(i.e. the fact that all probabilities are equal).

Our implementation of this algorithm consists of two phases. The first of these constructs the

arrays A and B and makes a first pass to consider all 2-interchanges of adjacent nodes i, i+ 1, making

improvements in the tour without updating the matrices. If an improvement is found, phase one is

repeated; otherwise, a second phase is initiated to check for other possible improvements until one

is found (upon which phase 1 is re-started). Thus most of the improvement is made in phase one,

and the algorithm runs in O(n2) on average.

8

Ill

2.4 The 2-interchange: Heterogeneous PTSP

Our testing procedures also include the 2-interchange heuristic in the case of non-homogeneous

probabilities. Here the formula is somewhat more complicated and computationally expensive. For

ease of illustration, we present the formula only for the special case where the interchanged nodes

are adjacent.

Consider an initial configuration and a final configuration differing only by the inversion of nodes

i and i + 1. Denoting by A(i, i + 1) the difference in cost of the two tours, we have

on-l i+r-1

A(i,i+ 1) = PiPi+l d(i,i+ r)pi+, J (1 - Pq)-r=2 q=i+2

n-2 i+r

EZd(i+li+ l + r)pi++l (1-pq)+r=1 q=i+2

n-2 i-r+l

E d(i, r)pi- r I (1-P,-r=l q=i-

n-1 i-1

Ed(i + 1,i+1-r)pi_,+l H (1-pq) (17)r=2 q=i-r+2

For the more general situation where the nodes are not adjacent, it is a straightforward but tedious

matter to generalize the homogeneous case. However, due to the absence of symmetry, there are

more interactions between inside and outside nodes. We illustrate this idea briefly.

Referring to Figure 1, we define a clockwise orientation to be movement along the circle clockwise

relative to the outside nodes (i.e. in the direction from j + 1 to i - 1 in the top picture), and we

define a corresponding counterclockwise orientation to be the opposite. Now we merely note that

from the point of view of the node i relative to the outside nodes, the only important difference is as

follows: moving in a clockwise fashion from i in I(i + 1, j - 1), the nodes j - 1,..., i + 1, j intervene

between node i and the outside nodes, while in I(i,j) the intervening nodes are j,j - 1,..., i + 1,

and the direction is counter-clockwise. Thus in I(i, j), the contribution to expected cost from the

interaction is

n-1

P, E d(i,i + r)pi+r (1 - p) +r-=k+l q=j+l

n-k-1 i-i

P[(1 - Pi+l)' (1-pj)] E d(i, i-r)pir H (1-Pq) (18)r=1 q=i-r+l

9

while in I(i + 1, j - 1) it is

n-1 i+r-i

Pi[(l-pi+l) (1-pj)] d(i, i + r)pi+r I (1-pq) +r=k+l q=j+l

n-k-1 i-1

Pi a d(i, i-r)pi-r II (1-pq). (19)r=l q=i-r+l

Subtracting (19) from (18), we obtain the net effect of the interchange with respect to this interaction.

The other parts of the formula are obtained in similar fashion.

2.5 The 1-shift heuristic: Homogeneous PTSP

In the realm of the TSP there has been some success with the so-called 3-interchange heuristic,

which could conceivably be adapted for the case of the PTSP. The idea is to improve an existing

tour by shifting an entire section (i.e. a set of nodes lying between two points) to a different location

within the tour. An initial ordering of

(1,..,i- 1,i,...,j,j + 1,...,k,k + ,...,n)

becomes, after an insertion of the points i,..., j between k and k + 1,

(1,...,i- 1,j+ 1,...,k,i,...,j,k + 1,...,n).

Unfortunately, the additional computational burden created by the probabilistic elements of the

objective function make the 3-interchange heuristic unattractive for the PTSP. However, Bertsimas

and Howell [6] have introduced the 1-shift, which is a degenerate (and much simpler) version of the

general 3-interchange procedure. This we now proceed to describe.

The 1-shift heuristic is obtained above when the set of points i,..., j consists only of a single

point i (hence the name 1-shift). The procedure differs fundamentally from the 2-interchange in that

there is no inversion of points on the tour. Bertsimas and Howell have argued that for small values

of p this structure-preserving property makes the 1-shift procedure more efficient. Their argument is

as follows. In the case of low p, any reasonable tour (e.g. one obtained via radial sorting) is likely to

be a good one already. Thus most inversions are likely to worsen rather than improve the objective

function. The 1-shift does not even consider such inversions and therefore saves computation time.

To derive the formula for the change in the objective value due to a 1-shift (see Figure 2), consider

an initial configuration

(1,...,i-,ii + ,...,j- 1,j,j + 1,...,n)

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i-i

i+2i+1 i+2

i j-ij-

- 1

Figure 2: Demonstration of the 1-shift procedure. At top is the configuration S(ij); below is the

configuration S(i,j - 1). The arrows indicate the original tour.

which becomes after a 1-shift of node of node i after j

(1,...,i- 1, i+ ,...,j- 1,j,i, + 1,...,n).

Denote the latter configuration by S(i, j) and its cost by C(i, j). Then we have the formula (for

details see [9])

C(i, j) = C(i,j - 1) +

p2

[(q-k q-(-))Ai,k+ 1 + (qk - qk-l)(Bi l- Bin-)+

(q - 1)(A/ 1 - Aj,n.-k) + (q-' _ 1)Bj,k+l +

(q,-1-k - qn-k)(Ail - Ai,k) + (qk-n - qk-(n-l))Bi,,,k+l +

(1 - q- 1 )Aj,.k+l + (1 - q)(Bj,1 - Bj,k+1)l , (20)

taking all indices modulo n.

As with the 2-interchange, the symmetry of the homogeneous problem simplifies the calculations.

2.6 The 1-shift: Heterogeneous PTSP

The 1-shift formula (20) can be extended to the heterogeneous case in the same manner as the

2-interchange procedure. We omit the straightforward but tedious derivation.

11

2.7 Sampling and Re-optimization

As mentioned at the outset, the goal of our investigation was to compare a priori heuristics with

strategies of re-optimization. Of course in practice exact re-optimization procedures are often not

computationally feasible. In our case, where the intent was to develop algorithms capable of being

run on small computers, exact methods proved impossible for problems of any reasonable size. The

set of possible outcomes to the PTSP has cardinality 2, so that even a fast machine would confront

a huge number of problems to be solved in order to obtain an expected cost. Thus in our studies

we were led to consider pseudo-reoptimization, employing (deterministic) heuristic procedures and

a sampling technique which we now describe.

Let us denote by Si a sample of points (generated from the n total points) representing locations

having a demand on a given realization i, i = 1,..., N where N is the number of samples taken.

Supposing for the moment that exact optimization were possible for each corresponding TSP, the

law of large numbers suggests that for large enough N,

E[PTSP] , $'--1 p(Sj)LTSP(Si) (21)

N

suggesting a lower bound for our a priori solution. The variance of the estimate so obtained can be

reduced through the use of a control variate procedure. To see this we write

E[ETSP] = E p(S)(LTSP(S) - L(S)) + p(S)L(S) (22)SCv sCV

The last term of this equation, which we shall denote as ', can be computed exactly once the

heuristic tour has been found. The first term, which we call t - t', must be computed using Monte

Carlo sampling. If we let s 2 denote the sample variance for the estimate of E[ETSP], we have that

s2 = Var(t - t' + 0') = Var(t) + Var(t') - 2Cov(t, t'). (23)

Since we expect that the terms t and t' will be strongly positively correlated, the technique should

(and does) improve our estimate's variance.

Before computational testing could begin, we confronted the question of what method to use in

solving the sampled TSP's. Since our test problems were on graphs of 100 nodes and our algorithms

intended for use on personal computers, heuristic procedures were necessary; nevertheless we wished

to achieve close to optimal results for purposes of comparison. A reasonable compromise between

tractability and accuracy was achieved by taking for each problem the minimum of two tours, one

obtained via SFC plus 3-interchange, the other obtained as the minimum cost tour of IS1/5 randomly

generated permutations of the (randomly generated) set S.

12

"I1

3 Computational Results for the PTSP-

In this section we briefly present some of the implementation details and then present and analyze

our computational findings.

3.1 Implementation Details

We implemented the heuristic and sampling algorithms in a package written for the Apple Macintosh.

This package also contains algorithms for the TSP, k-TSP, k-PTSP, VRP, and PVRP. For this

application we used the C programming language and also made extensive use of the graphics

capabilities of the Macintosh Toolbox (for menus, graphics, etc.) We present here only some of the

more important details.

We begin with the user inputs. Principally, these are the number of points n for which the PTSP

is to be solved, the common probability p (in the case of a homogenous problem), and the type

of heuristic (SFC or radial sort) to be used in constructing initial TSP solutions. If heterogeneous

probabilities are specified instead, these are generated randomly on the interval [0, 1] and stored

as a vector of length n. The locations of the points (cities) are specified as n-vectors of z- and

y-coordinates, either directly by the user, or as randomly located in the unit square. Distances are

stored in a matrix, though space considerations in larger problems may warrant more sophisticated

methods. All tours constructed and modified in the course of a solution are maintained in the

following way. At any given time, the current tour is stored as a permutation of the integers

1,...,n. This permutation acts as an index for the coordinate vectors. Thus z[tour[i]] gives the

z-coordinate of the ith point on the current tour.

After the input of the initial parameters, the program calculates and displays an initial TSP

tour through all points, using the heuristic specified earlier. With this initial tour as a basis, the

user is then free to try either of the local improvement techniques to modify the tour. As the tour

is improved, the display on the screen shows the changes and the percentage improvement in the

objective function. There is also a batch option available to specify a series of problems (e.g. random

problems) to be solved. The sampling/re-optimization scheme is also run under this option.

3.2 The Testing Procedure

Our testing procedure was as follows. In the homogeneous case, we generated five problems (n =

100) for each of the common probabilities p = 0.1,..., 0.9. By a problem we mean a set of n random

13

locations in the unit square. Each problem was then solved by each of the four combinations of

heuristics (space-filling curve plus 2-interchange, space-filling curve plus 1-shift, radial sort plus 2-

interchange, and radial sort plus 1-shift) in addition to the re-optimization scheme. In the latter

case, for each problem, 50 instances (i.e. sets of nodes S) were generated and the corresponding

TSP solved to near-optimality. The five problems were different for different probabilities (i.e. there

were 45 separate problem-probability combinations), but in a separate testing procedure, the same

five problems were used across all values of p, in order to see the effect of p on solution values.

In the case of the heterogeneous PTSP, we generated five original problems. For each of these we

generated n random probabilities P,..., p, and solved with the same four combinations of heuristics.

For the re-optimization, 50 instances of the TSP were sampled for each problem, with each point i

appearing in the corresponding TSP according to its probability pi. We turn now to a consideration

of the results.

3.3 Conclusions

The homogeneous PTSP

Tables 1 and 2, together with Figure 3, summarize the chief computational test results for the

heuristics we have described. We have not included running time here but choose instead to make

some qualitative remarks in this regard. The data consist of 5-problem averages of the objective

value attained by four diffferent heuristics and re-optimization. Table 1 refers to the homogeneous

case, with the common probability ranging from 0.1 up to 0.9. As is evident from the figure,

all heuristics, with the sole exception of radial sort + 1-shift, give roughly the same performance.

Moreover, they are also close in performance to the re-optimization heuristic, typically within 1%.

Since we suspect that the latter gives results which are quite close to optimal, the data suggest

that the a priori tours constructed have low expected cost, an observation which has considerable

practical significance. As noted at the outset, the a priori approach demands far fewer resources than

re-optimization and establishes a uniformity of service which re-optimization cannot provide. All of

this makes a priori solutions far more easy to implement, so the additional fact that computational

experiments suggest good cost performance is an encouraging fact.

The worst performing heuristic appears to be radial sort + 1-shift. Re-optimization costs are

usually better than those for the a priori heuristics, but the differences are not great. However, radial

sort + 1-shift gives consistently higher cost, ranging all the way up to a 14% difference at the higher

values of p. The explanation for this relatively poor performance probably has more to do with the

14

III

P Objective Value (avg. of 5)

Rad + lshft Rad + 2opt SFC + lshft SFC + 2opt Re-opt.

0.1 2.9848 2.9806 2.9921 2.9909 2.9844

0.2 4.1539 4.1173 4.1813 4.1847 4.1262

0.3 5.2879 5.0383 5.0644 5.0311 5.1035

0.4 6.4961 5.8413 5.9410 5.8886 5.7742

0.5 7.2901 6.6037 6.7231 6.5663 6.3741

0.6 7.8895 6.8543 7.1218 6.8726 7.1032

0.7 8.5207 7.4124 7.5276 7.4433 7.4028

0.8 8.6378 7.8531 7.7075 7.6216 7.5826

0.9 8.8538 8.2652 8.3171 8.3499 8.0551

Table 1: Heuristics vs Re-optimization: Homogeneous PTSP

Heuristics v. Re-optimizatlon

.0 0.2 0.4 0.6 0.8

-I Rad + Ishft

- Rad + 2opt-I SFC + Ishft-e SFC 2opt_ Re-opt

4 Asymptote

I .0

p

Figure 3: Heuristics vs. Re-optimization for the Homogeneous PTSP

15

10-

8-

0 6-o

2-0

U - I - I~~~~~~~~~~~~~~~~r r r I

1-shift procedure than with radial sort. As noted earlier, the radial sort has good properties for

low values of p, where the coupling effects of node-interactions become relatively more important.

This idea is confirmed by the table entries for p = 0.1,0.2. At higher values of p, however, the

importance of the coupling effect diminishes, and in general the radial sort is inferior to SFC. More

important than this, however, are the characteristics of the 1-shift procedure. The 1-shift procedure

tends to preserve original order, while 2-interchange attempts to make numerous inversions. Our

intuition is that in considering these inversions, 2-interchange examines a wider range of potential

improvements, despite the fact that the 1-shift procedure actually operates over a larger space. By

considering more radical alterations in the shape of the design, 2-interchange succeeds in achieving

a lower cost overall.

We believe that the choice of radial sort versus space-filling curve is of relatively little consequence

as long as the subsequent improvements considered (via local optimization) are extensive enough. In

the case of 1-shift, the difficulty arises because the original shape is not exposed to a wide enough

series of possible transformations.

With regard to running time, all heuristics for the homogenous PTSP were quite fast, with

the 2-opt procedures considerably faster (by a factor 2 or 3) than their 1-shift counterparts. The

algorithms were easily implemented on small personal computers - in our case, the running time for

a single 2-opt problem of 100 nodes (with p = 0.7, say) was on the order of 2-3 minutes (increasing,

of course, in the value of p) on a Macintosh II. To run the entire set of 45 problems (with p ranging

from 0.1 to 0.9) took about 1 1/2 hours at the shortest (SFC + 2opt) and 4 hours at the longest

(radial sort + 1-shift). Taken together with the cost data of the table, this information suggests

that in the homogeneous case, a combination of the spacefilling curve and 2-interchange heuristics

quickly produces tours which have close to optimal expected cost. From a practical standpoint, this

combination gives the best performance.

Also included in the figure is a curve indicating the asymptotic expected cost of the PTSP, which

for large n is

Cn ~ 3TSPV,

where PTSP T 0.72. We see that for n = 100, asymptotic cost is not yet achieved, although the

actual cost does follow roughly the same trajectory as the asymptotic cost over the range of p.

The heterogeneous PTSP

The heterogeneous case differs in several respects from the homogeneous one. Computation times

for the heterogeneous case were significantly higher than for the homogeneous case, with all a

16

III

Objective Value (avg. of 5)

Rad + shft Rad + 2opt SFC + lshft SFC + 2opt Re-opt.

6.2280 6.5111 6.0413 6.5971 6.9315

Table 2: Heuristics vs Re-optimization: Heterogeneous PTSP

priori heuristics requiring over twelve hours to finish (5 problems). In fact, in the case of the 2-

opt procedures, run-times were so excessive that we chose instead to impose a truncation scheme

on the number of allowable iterations (hence the higher costs for the combinations employing 2-

opt). The reasons why 2-opt is slow for heterogeneous demands while it is fast for homogeneous

demands are clear: the computational burden of computing the expected costs (which must be

evaluated to compare local alternatives) is substantially higher in the heterogeneous case. The

practical implications of the loss in speed are not too severe, however, since the truncated algorithms

(which run very quickly - five problems in 23 minutes) give reasonably good answers.

A noticeable phenomenon in the heterogeneous case is the superiority of all a priori procedures

over heuristic re-optimization, by a margin of about 8% in the case of the 1-shift procedures. This

finding is in contrast to the homogeneous case, where the heuristic re-optimization procedures dif-

fered only slightly from the others. There is some explanation for this finding. The heuristics used

in the re-optimization tend to perform well when the demand points are uniformly distributed, but

their performance deteriorates when there is significant clustering. While homogeneous probabili-

ties tend to give a uniform distribution of points with positive demand, in the heterogeneous case,

clustering is more likely. Moreover, the non-reoptimization heuristics are fairly sophisticated in that

they take careful account of the interactions between nodes (i.e. they explicitly consider the effect

that each node's presence or absence has on the others in the tour).

While the heterogeneous case is certainly more difficult, our conclusion is that the truncated

versions of the heuristics perform well, both in terms of run time and objective value. With their

attractive simplicity, they should prove useful in practice.

17

4 Heuristics for the 1-Vehicle Stochastic Routing Problem

From the PTSP we now turn to the 1-vehicle PVRP, for which we propose a heuristic we call the

cyclic heuristic. After presenting some worst-case analysis for this heuristic, we introduce some

refinements (local improvement procedures and dynamic programming) and perform computational

experiments.

4.1 The Cyclic Heuristic

The cyclic heuristic develops a visitation strategy for the 1-vehicle PVRP based on a TSP tour of the

n demand points. By visitation strategy we mean an a priori plan for visiting the demand points.

Given any route on the n demand points, a visitation strategy is obtained by starting at the depot

and visiting points as specified by the route, returning to the depot whenever capacity is reached.

Under strategy a, all points are visited, whereas for strategy b, wherever a zero demand is realized,

it is skipped. Most of our analysis restricts attention to strategy a; some remarks at the end of the

section address strategy b.

A simple example illustrates the idea of a visitation strategy. Suppose the TSP solution is 2, 4, 3, 1

and the vehicle reaches its capacity in the course of fulfilling demand at point 3. In this case, it

returns to the depot, empties its load, and resumes its tour at point 3. If the remaining demand at

3 plus the demand at 4 does not exceed capacity, the final route is 0-2-4-3-0-3-1-0.

Note that by considering each of the n points of the tour as a starting point (i.e. the first point

to visit after leaving the depot), we obtain n potential visitation strategies. The cyclic heuristic

works by considering each of these n possibilities and selecting the one with the best expected cost,

which is given by the expression

~~n n

E[Lr] = (1 -i)d(id i + 1) + (2d(i, 0) + i[d(i, 0) + d(i + 1, 0)]), (24)i=O i=1

assuming that the underlying tour is 1, ... , n. This formula is merely that of equation (10) re-written

by expanding the terms s(i, i + 1).

Prior to computational testing, we assess the worst-case performance of the cyclic heuristic and

examine its asymptotic behavior relative to a strategy of re-optimization. The following are the

two major results, which hold under the so-called visitation restriction, the requirement that under

re-optimization, all locations must still be visited (as in strategy a). We employ the terminology

L, _ (random) cost of cyclic heuristic solution under strategy a

18

III

Ra = (random) cost of re-optimization solution under strategy an

- Zd(O, i)/ni=l1

3 SP = BHH constant for Euclidean TSP - 0.72.

Theorem 1 For strategy a under the visitation restriction,

E[L] < 2. (25)

Theorem 2 Let X 1, ... , Xn be a vector of i.i. d. demands, whose locations (together with the location

of the depot) are independently and uniformly distributed throughout the unit square. Then for a

vehicle with capacity Qn under strategy a,

(i)If hi; - 0 as n -, oo then

lim QnE[LI ] = 2E(X)? (26)n-oo n

(ii)If Qn - oo as n - oo then

lim E[L]= /TSP (27)n-oo /

Similarly, for strategy b with p = Pr{zero demand},

(iii)If - -- O as n - oo then

lim QnE[L] = 2E(X)r (28)n-oo n

(iv)If Qn - oo as n - oo then

lim E[a = d(p) (29)n-floo/-

where *Tspv/ < ,3(p) < 0.92V/.

For proofs of the above theorems see Bertsimas [7].

Note that under strategy a, the above shows that the cyclic heuristic is asymptotically optimal

if the initial tour is the optimal TST. The heuristic is also asymptotically optimal for strategy b in

the case where limnO 0. = 0.

In practice, of course, the optimal TST is not available; instead, we must begin with a heuristic

solution. However, there are several practical refinements which can be made to the cyclic heuristic.

We describe these next.

19

4.2 Local Improvement Procedures

In the case where the node demands are independent and identically distributed (what we have called

the homogeneous PVRP), the initial TSP tour can be refined via local improvement procedures.

These rely on the fact that with a homogeneous PVRP, the return probabilities 7i and i depend

only on the order (and not the identity) of the nodes. Although the procedure to be described

here could be adapted for strategy b, the details are quite tedious and the computation expensive.

Therefore, as we noted above, we consider only the case of strategy a here.

The local improvement algorithm is a 2-interchange procedure similar to those used for the PTSP

in the previous section. It uses the expected cost (24) as the objective value, considering at each step

the potential improvement which would be obtained upon inverting a section of the tour between a

current pair of nodes. For example, if the pair is (i,j), an initial ordering of

(1,...,i-,i,i+ ,...,j-l,j,j+l,...,n)

would become

(1,...,i- 1,j,j- 1,...,i + 1, i,j + 1,...,n).

Again we shall distinguish between outside and inside nodes. In light of the previous remarks for

the homogeneous PVRP, the contribution of the outside nodes 1,. .. , i - 1 and j + 1, ... , n in (24)

is unchanged. For the inside nodes, there is a change 6 = - i, where

i j j-1

6,b= (1- y)d(k,k + 1)+ (26 + )d(k,0)+ i - d( + 1,0)k=i-1 k=i k=i-1

b! = (1- 7 i- 1 )d(i - 1,j) + (1 - yj)d(i,j + 1) + Z(2i+j k d(k, 0)) +k=i

j-1 j-1

Z(1- 7i+j-kl)d(k,k + 1)+ E ?1i+j-k- 2 d(k + 1,0).k=i k=i-1

The derivation of these expressions is straightforward bookkeeping, given the definitions of the break

probabilities 6 and y. Local improvement proceeds by making all interchanges where 6 > 0.

Local improvement may be incorporated into the cyclic heuristic in two ways. Most simply, it

may be used to improve the initial TSP tour, resulting in a new initial tour for the cyclic heuristic.

Alternatively, local improvement may be attempted for each cyclic permutation of the original tour

(this obviously is more expensive computationally). The test results to follow shortly reflect this

second, more expensive implementation. In either case, subsequent improvement in the cost function

is possible via the dynamic programming procedures introduced next.

20

III

4.3 Dynamic Programming Procedures

All procedures described thus far are static in the sense that information obtained en route (i.e. by

the driver) does not affect decisions. Incorporating such information simply is the essence of the

dynamic programming procedure described next. Suppose then that we have an initial tour, say

a good TSP tour, on the demand points. We can then describe a dynamic programming problem

in which the decision points occur each time the vehicle departs a demand node i carrying a given

load k. We define C(i, k) to be the expected remaining distance to travel, given that the optimal

strategy is followed, when the vehicle has just visited node i - 1 and is now carrying a load k. At

each decision point, the choice is either to continue to the next point i (and risk the possibility that

the additional demand there will exceed the remaining capacity Q - k), or to return to the depot

to drop off the current load before proceeding to node i. Because the triangle inequality is assumed

to hold throughout, it will sometimes be better for the vehicle to return to the depot only partially

full, since there may be substantial probability that otherwise the vehicle will become full before

filling all of the demand at the next location (necessitating a return trip to the depot and a second

trip out to fill the remaining demand). Such a cautious strategy will especially pay off in the case

where the next location is quite distant from the depot and thus quite expensive to visit more than

once.

We may formally write the dynamic program as follows. Define the maximum demand value for

node i to be K(i)[K(i) < Q Vi]. Then noting that PI = 0 for j > K(i), we have

Initialization:

C(n +1,) = d(0,n)

Recursion (for i = n,. . . , 2):

QC(i,k) = min d(, i - 1) + d(O,i) + E C(i + 1,j)p,

j=oQ-k

d(i- 1,i)+ E C(i + 1,k + j)pi +j=o

Q '

Z2> p p(2d(0, i)+ C(i + 1, + j - Q))j=Q-k+l

and

k*(i-1) =

21

min k C(i, k) = d(O, i- 1) + d(O, i) + Z C(i+ 1,j)pi .I inj= 1~~~~j=1

The first three terms within the curly brackets reflect the strategy of returning to the depot first.

This cost consists of the distance from the present point to the depot (d(O, i)), the distance from

the depot to the next point on the tour (d(O, i + 1)), and the expected remaining cost entailed in

proceeding with the tour from that point. The second group of three terms reflects the strategy of

continuing to the next demand point. This cost consists of the direct distance-cost of proceeding to

the next point, (d(i - 1, i)), plus the expected remaining cost entailed in continuing from there. This

in turn is divided into two summations, reflecting the two possibilities of exceeding or not exceeding

capacity at the next point. The former possibility incurs a cost of 2(d(0, i)) of a return to the depot

necessitated by its full load. The term k*(i) is what we have called the threshold value for node i

and has a very straightforward interpretation: if the load after visiting node i is at least k'(i), it

is optimal to return to the depot before filling further demands. These threshold values, together

with the underlying order of visitation, constitute a solution for any given starting point, and the

expected total cost is simply =_o pC(2,j) + d(O, 1).

Note that the dynamic program runs in O(nK2 ) time and requires O(nK) storage capability,

where K = max{K(i)}. Unlike the local interchange argument described above, the dynamic

programming procedure does not require that demands be identically distributed.

4.4 The Full Cyclic Heuristic

There is an important difference in philosophy between local improvement and the dynamic pro-

gramming approach: the former assumes that the vehicle only returns to the depot when full, while

the latter allows the vehicle to return partially full. It is not clear on theoretical grounds whether

together the two procedures will always produce a better solution than that which would be obtained

by dynamic programming alone (starting with just an initial TSP tour). On the other hand, the

dynamic program can only improve upon a local optimization solution. An important question is

whether the local improvement substantially improves cost, since its inclusion increases run time.

Assuming for the moment that we include local optimization at each of the n steps, we have the

full cyclic heuristic as follows:

Algorithm for strategy a:

22

III

1. For demand points 1... n compute a solution to the TSP using a set of good heuristic proce-

dures.

2. For each of the n orderings implied by the cyclic permutations of the above tour, use local

optimization to obtain a new tour. Then apply the dynamic program to both the original tour

and the improved tour, obtaining threshold values in each case.

3. Choose the lowest cost visitation strategy of these 2n possibilities.

4. Output the final tour, its expected length, and the threshold values.

We remark that the full cyclic heuristic is technically not polynomial time because the local im-

provement procedures are exponential in the worst case. A simpler version of the heuristic (omitting

local improvement) has complexity O(n2K2).

4.5 Remarks on Case b

Although we have not presented any detailed analysis, the cyclic heuristic may of course be adapted

for case b, where zero demands are skipped. Local improvement procedures based upon expected cost

are again available, though they are more computationally demanding. The dynamic programming

procedure is not adaptable to case b: the computation of the threshold values is only possible a priori

because we know that all nodes must be visited. Under strategy b, we obtain prior information that

some nodes have zero demand, but incorporating this information into the tour design amounts

to pseudo-reoptimization. While our computational testing is solely for the case of strategy a, we

suspect that the algorithm will also work well when adapted for case b.

5 Computational Testing for the Cyclic Heuristic

5.1 Implementing the Cyclic Heuristic in 'C'

We describe one implementation of the cyclic heuristic under strategy a. This implementation is

the one used for all of the subsequent testing, which was performed on a DEC 3100 Vax work-

station computer. As noted earlier, the original PVRP program was part of a larger package for

Macintosh; unlike in the case of the PTSP, however, the running times (especially for heterogeneous

demands) and memory requirements exceeded the capabilities of small Macintosh machines. How-

ever, a significant part of this complexity arose from the need to compare different features (e.g.

23

local improvement versus dynamic programming). In practical terms, this means that the more

streamlined versions of the algorithm (e.g. dynamic programming with no local improvement) are

implementable for smaller machines. As mentioned earlier, the complexity of the most streamlined

version is O(n2 K 2 ). In what follows, we present some of the more important implementation details.

The program begins with the input of the necessary parameters: the number of nodes n, the

vehicle capacity Q, and the type of demand distribution. The type of distribution, e.g. normal, is

assumed to be the same for all demand points, though the parameters, e.g. the means and variances,

may vary. Our program handles two types of distributions: a discretized normal distribution and a

discrete uniform distribution. In the former case, the user specifies a mean and standard deviation

for the demand at each node, and the discrete probabilities are calculated on the basis of a standard

normal table with a suitable rounding procedure. For the more simple discrete uniform case, the

user merely specifies the range for each node demand.

Given these parameters, the first step in the program is the calculation of the probabilities i

and i via the procedure described in section (1). If the demands are specified to be identically

distributed, this calculation need only be done once, since in this case, a change in the order of the

points merely permutes the values of the yi's and 6i's. In the case of non-identical demands, local

improvement is not used, and the only reason for calculating the probabilities is the calculation of

the expected cost without dynamic programming (the dynamic programming procedure calculates

its own expected cost as part of the recursion - hence our remarks above on streamlining the

procedure by omitting local optimization).

The main part of the program proceeds as follows. For each of a (user-specified) number of

problems, the user may specify the locations of n points in the unit square, or these points may

be taken to be independently uniformly distributed. As with the PTSP, x- and y- coordinates

for points 1,..., n are stored as n-vectors, and inter-node distances are computed and stored as a

matrix. An initial TSP solution is constructed via SFC and improved via a combination of 2- and

3-interchange procedures. The traveling salesman tour so obtained is usually with 2-3% of optimal.

The principal inner loop of the program consists of permuting this initial tour by one place each

time. For each permutation, local-optimization is performed (if specified so by the user), and the

dynamic program is run to calculate the threshold values. For the dynamic programming, we use a

cost matrix C, where C[tour[i]][k] denotes the expected cost remaining after departing the i - th

node on the tour carrying a load of k. The algorithm works backward from the nth node in the

usual fashion. Thus at each of the n iterations, we calculate both an expected cost without dynamic

24

11

programming (c.f. eq.(24)) and an expected cost incorporating the threshold values. The best cost

value is updated over the course of the loop, and the corresponding tour saved. The output consists

of the two tours obtained plus the threshold values for the second tour. For the Macintosh version,

graphic displays are available.

5.2 Strategies of Re-optimization

As in the case of the PTSP testing procedures, the ideal candidate for comparison is a strategy of

re-optimization in which the (optimal) exact routes are formulated once the demands are known.

To obtain a full comparison, one would generate all instances of demand for the given demand

distribution(s), solve each instance optimally, and calculate an overall expected cost

E[L']= E P(il .... in)L'(il,..., i.), (30)il ,.,iu,

where L(il,..., i,) is the optimal vehicle routing cost for the demand realization (X1 = il,..., X, =

in) and P(-) denotes the (discrete) joint demand distribution on n nodes.

Even more so than in the PTSP case, there are prohibitive computational problems with such

an approach. The number of terms in the summation in (30) is enormous for all but very small

problems (for example, if the demands are independently discrete uniform on [1, 10], the total number

of VRP's to solve is 10n). To overcome this problem of dimensionality, we again employ sampling

ideas. Instead of considering all instances of demand, we generate a sample of the demands from

the underlying distribution and solve these instances optimally. Each sample yields solution with

cost L*(i,... ,i), with k indexing demand samples and ranging from 1 up to S. An estimate of

the overall expected cost in following a re-optimization strategy is then

S

Lr T/~ k1(31)S = S E L ,.).,in) (31)k=1

While this approach eliminates the problem of the large state space, the estimation implicit in (31)

is still quite computationally intensive because of the requirement that each sample VRP be solved

to optimality. For small n, exact solutions to the VRP can be obtained easily, but problems bigger

than 15 nodes quickly become intractable. For example, a dynamic programming formulation of

the problem in [10], in the absence of additional side constraints, essentially amounts to a complete

enumeration of all subsets of nodes (examining each subset as a potential subtour), an enumeration

of 2" sets. Thus, in order to conduct tests on problems of reasonable size, we are forced to abandon

true "re-optimization" as the standard for comparison.

25

�1_�X 1_Y�^__)______X^_�I�____

With exact optimization possessing these drawbacks, it is sensible to consider instead good

heuristic solutions. Following this idea, for each sampled demand instance we solve the corresponding

VRP by a (deterministic) heuristic to obtain a cost Lh(il,...,in). For S samples we obtain the

estimate

=~ E h(ik, ik(32)k=l

This is the approach we have employed in our testing.

We implemented two heuristics for the VRP: a clustering heuristic due to Christofides, Mingozzi,

and Toth [10] and the well-known Clarke-Wright savings algorithm [11]. The clustering heuristic

groups nodes into clusters according to a local cost criterion, then constructs subtours through

these clusters using heuristic TSP procedures. Particular nodes are chosen as initial seed points,

and the remaining nodes are assigned according to this measure of relative cost, thereby forming a

cluster. The local cost criterion may be simple distance to the seed, average distance to all points in

the existing cluster, or some other suitable measure. Clusters are formed both sequentially and in

parallel, with the better solution selected. The program requires a somewhat extensive use of data

structures. The Clarke-Wright procedure begins with each demand point on a separate subtour, then

consolidates these separate tours according to the potential savings realized by such consolidation.

Both algorithms were encoded on the assumption that all points (including zero-demand points)

were to be visited, in keeping with our testing of strategy a.

5.3 The Testing Procedure

Although our code was originally intended for the MacIntosh (to take advantage of the graphics

package available there), for the purposes of testing we uploaded onto a VAX 3100 Workstation

using the more standard version of C. This change was necessary in order to obtain results for larger

problems (greater than 20 nodes) in a reasonable amount of time. However, as noted before, simpler

versions of the algorithm (i.e. those not performing local optimization and not calculating the return

probabilities) have shorter run-times.

The testing format was as follows. For each type of PVRP (a type corresponding to particular

values for n, Q, and the demand distribution), 20 test problems were constructed by randomly

choosing n points in the unit square. In the case of the re-optimization heuristics, for each of these

test problems 100 demand instances were simulated according to the specified distributions. We

obtained estimated re-optimization objective values by averaging over instances for each problem,

26

n Q Distribution Obj. Value (avg. of 20)

Cyclic Cluster ClarkeWright

20 40 N(5,1.66) 4.851 5.029 4.835

20 40 U[1, 10] 5.117 5.24 4.99

20 40 N(10, 3.33) 6.675 6.893 6.582

20 40 U[0, 20] 6.803 7.037 6.585

20 40 N' 9.372 9.904 9.623

20 40 N 9.912 10.545 10.278

30 40 N(10, 3.33) 9.273 9.684 9.140

30 40 U[0, 20] 9.416 10.045 9.146

40 50 N(12,4) 11.390 12.085 11.210

40 50 U[0, 24] 11.571 12.763 11.245

50 50 N(5,1.66) 8.396 9.016 8.376

50 50 U[1,10] 9.099 9.473 8.686

50 50 N(12,4) 13.999 14.834 13.701

50 50 U[0, 24] 14.097 15.775 13.706

50 50 N' 18.377 20.056 18.708

50 50 N* 19.56 22.22 20.676

100 30 U[1,20] 33.724 39.101 34.701

Table 3: Cyclic Heuristic vs. Re-optimization

and we then

obtained for

obtained

expected

an overall average for the 20 problems. For the cyclic heuristic, values were

costs under four scenarios: no dynamic program and no local optimization

(basic heuristic), no dynamic programming with local optimization, dynamic programming without

local optimization, and dynamic programming with local optimization.

5.4 Discussion

We turn now to a discussion of our computational results. There are a number of issues to address:

* How does the cyclic heuristic compare to re-optimization?

* At what problem size does asymptotic behavior emerge?

27

. . A _

* What are the relative effects of local improvement and dynamic programming procedures?

* What are the relevant conclusions for the practitioner?

Consider first the comparison of the cyclic heuristic to re-optimization. Our expectation prior to

testing was that the cyclic heuristic would perform quite well, considerably better than its worst-

case bound of two. Our test results against heuristic re-optimization procedures do suggest that

the a priori approach works well. Consider the data of Table 3. Here we have classified problems

according to n, Q, and demand distribution. With respect to the latter category, N(p, a) denotes

a discretized normal distribution whose continuous version has mean p and standard deviation a,

while U[a, b] denotes a discrete uniform distribution on the interval [a, b]. These two distributions

correspond to identically distributed demands. We have also included two other distributions, N;

and N2, which denote independent but non-identical normal demands at the n nodes (i.e. normal

demands of different means and variances). For example, the distributions N; are a repeating cycle

of N(10, 3.33), N(16,5.33), N(26,8.66), N(20,6.66), N(14,4,66), N(24,8), N(18,6), N(12,4), and

N(22, 7.33) (the cycle repeats every nine nodes). The distributions N2 are similar but have wider

variation between nodes.

Figure 4 gives a graphical summary of the data in the table, showing the objective values for the

three different approaches (Clarke-Wright, cluster heuristic, full cyclic heuristic) for the first 16 test

problems of the table. From the data, we see that the cyclic heuristic performs very impressively in

comparison to re-optimization. In fact, the heuristic is consistently superior to the cluster scheme,

giving a lower objective value for all problems considered, the margin of superiority ranging from

around 2% for the smaller test problems up to 16% for the 100-node problem. This result agrees

with the finding that vehicle routing heuristics based upon cluster first - route second schemes can

be asymptotically sub-optimal. This claim has been demonstrated in the case of the well-known

sweep heuristic [12], and it seems also to be the case here, although our results do not cover large

"asymptotic" cases. But even for the smallest problems we consider, the cyclic heuristic dominates

the cluster procedure and does so for a number of choices of the local cost criterion.

Clarke-Wright does manage to out-perform the cyclic heuristic for problems smaller than 50

nodes, but the margin of superiority is never more than 3%. Moreover, in all cases where demands

are non-identical, the cyclic heuristic outperforms both of the re-optimization schemes. While we

certainly are not surprised to see behavior which departs from worst-case expectations, the results

here suggest that Clarke-Wright does not perform well under certain conditions of widely- varying

demand, quite probably due to clustering effects. It is also clear that for larger problems, the cyclic

28

Cyclic heuristic v. re-optimization

-a- Cyclic-- Cluster-- Clark-Wright

- ,-u.u-E-s-I�,.,-I.I-I-,�u-..I. I

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Problem # (from table)

Figure 4: The heuristic vs. re-optimization

29

25'22'212019181716-

> 15-_ 14-

1312-

8-7-6-5-A.

A--

- .I-III-I-I-I-II-III-I-I-I-III-I

Asymptotic Behavior of Cost

r D)/Qial cost

100

No. of nodes

200

Figure 5: Comparing asymptotic and actual objective values

heuristic out-performs both re-optimization schemes. This phenomenon reflects the asymptotic

optimality of the cyclic heuristic, a characteristic which is not shared by the others. Our results

show that the margin of superiority of Clarke-Wright shrinks as the number of nodes increases, with

the cyclic heuristic eventually gaining the edge. For example, in the 100-node test problem, the cyclic

heuristic is better by 3%. We also note here briefly that the greater variance of the uniform demand

distribution increases cost (relative to a normal distribution with the same mean), a phenomenon

which is not surprising.

Asymptotic behavior of the heuristic is another matter to consider. Figure 5 illustrates one

example. Here we have plotted the objective values obtained for single test problems with different

values of n. For all of these problems, Q = 30 and the demand distribution is U[1, 20]. Included also

are the values of 2nE[D] / Q, the expression for the asymptotic cost when capacity is held constant.

We see here quite clearly the rough linearity of the objective value under these circumstances. The

objective value and the asymptotic value differ by a roughly constant amount up to n = 150, the

30

50 -

40 -

30

20 -

10-

0*0

I I=

n Q Distribution Obj. Value (avg. of 20)

LI,DP no LI,DP LI,no DP no LI,no DP

20 40 N(5,1.66) 4.851 4.884 5.255 5.295

20 40 U[1, 10] 5.117 5.142 5.461 5.497

20 40 N(10,3.33) 6.675 6.720 7.147 7.204

20 40 U[0, 20] 6.803 6.830 7.236 7.262

30 40 N(10,3.33) 9.273 9.376 9.820 9.944

30 40 U[0, 20] 9.416 9.479 9.900 9.976

40 50 N(12,4) 11.390 11.473 11.979 12.101

40 50 U[0,24] 11.571 11.615 12.084 12.171

50 50 N(12,4) 13.926 13.999 14.558 14.648

50 50 U[0, 24] 14.097 14.152 14.647 14.712

50 50 N(5,1.66) 8.396 8.580 8.908 9.118

50 50 U[1, 10] 8.998 9.099 9.536 9.646

Table 4: Effects of Local Improvement Procedures and Dynamic Programming

largest size considered. Naturally we would expect this difference to diminish to 0 as the number

of nodes grows very large relative to capacity. Additional runs (with smaller capacity) not shown

here do indeed confirm this result. They also confirm that when capacity grows faster than v/i, cost

grows at the rate of O(V').

What about the different effects of local optimization and dynamic programming? Here the

results, summarized in Table 4 and Figure 6 are perhaps of greater practical importance. We compare

the costs attained under the four different combinations of the local improvement and dynamic

programming refinements. The data show that the improvement due to dynamic programming is

roughly 6% for problems smaller than 50 nodes, while improvement due to local optimization is

less than 1%. As the number of nodes increases, these improvements decrease to 0, in keeping

with the asymptotic optimality of the basic heuristic. Our experience shows that the inclusion of

local optimization procedures increases running time by a factor of 2 to 3. The dynamic program

is comparable to local improvement in its run-time performance. It is important to note that in

order to evaluate expected cost, the algorithm must either explicitly calculate the probabilities

and 6, or it must perform the dynamic programming enumeration (either operation being O(nK2 )).

Eliminating the local procedures makes the algorithm pseudopolynomial (O(n2 K 2 )) in the worst

31

---n�-�v�u���---·-·-------··-··���--·---

Local Improvement v. Dynamic Programming

onlynly

I - I

4 6I _ _ ~~ ~ ~ ~~ ~ ~ ~~~~~~~~ I I I

8 10 12 14

Problem # (from table)

Figure 6: Effects of local improvement and dynamic programming

32

16

14-

12-

O

8-

6-

4.0 2

I

case and does not substantially worsen cost performance, while eliminating dynamic programming

gives a 5 to 10% deterioration for problems of small to moderate size. Since the expected costs must

be calculated in any event, retention of the DP procedures (where practical) is recommended.

6 Conclusions

The probabilistic versions of the TSP and VRP are clearly important in a large number of practi-

cal situations. Especially in cases where the supplier does not possess an extensive capability for

predicting demand, uncertainties will have a significant impact on operations strategy. Yet it is

precisely in these situations where the supplier is also unlikely to have the capability or the desire to

re-solve routing problems on a very frequent basis. Instead, it is likely that she will prefer a simple

strategy which can be implemented easily. What our results indicate is that simple a priori strate-

gies are robust, in the sense that they perform well over a wide range of actual (realized) demands.

Moreover, our proposed approaches require only infrequent computational effort, strongly increasing

the chances for successful implementation. As noted in [1], ease of implementation may strongly

depend upon this need for solving the problem only periodically.

For the practitioner, our results suggest that very simple procedures (SFC + 2-opt) will give ad-

equate results for the PTSP, while the cyclic heuristic performs well for the PTSP. These procedures

require very modest computational resources and give very simple strategies. In addition, the latter

can be improved somewhat by incorporating the dynamic programming procedure, which requires

some further implementation in terms of enacting rules for vehicles (when to make returns to the

depot).

More generally, our results show that stochastic vehicle routing problems are amenable to fairly

simple solution techniques based upon traveling salesman tours. We have shown that these proce-

dures are in fact quite robust to the underlying stochasticity in demand.

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